Exotic embedded surfaces and involutions from Real Seiberg-Witten theory

Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on $\mathbb{CP}^2$ and infinitely many exotic smooth embeddings of $\mathbb{RP}^2$ in $S^4$. In this paper we extend Miyazawa's construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in 4-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg--Witten theory. We study Miyazawa's invariant, relate it to the Real Seiberg--Witten invariants of Tian--Wang and prove various fundamental results concerning the Real Seiberg--Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg--Witten invariants, a connected sum formula and a fibre sum formula.